dimetris. the theorem of thales

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58 Dimitris Patsopoulos and Tasos Patronis

The International Journal for the History of Mathematics Education

Geometrical Education and Textbooks in the 19th Century

National public systems of education emerged during the 19th century, due atleast in part to the the impact of the French Revolution. One important feature ofthese changes was the establishment ofsecondary education, a level of educationbetween elementary education and higher education at universities. One of the

results of the pedagogical and didactical demands of this century was that largechanges took place in mathematical education, and particularly in the teaching ofgeometry (Cajori, 1910; Schubring, 1996, 1985). The subject of geometry becameestablished in lower levels of secondary education, and in primary education.Also, the method of teaching geometry underwent radical change in seniorgrades of secondary education, frequently becoming (as in England, France, andsomewhat less in Germany) an exercise in logic (Smith 1900, 303) within thecontext of introducing students to the art of syllogism.

Substantial changes also took place in the system of how textbooks wereauthored and brought into circulation. Many new authors appeared, particularly

in secondary education, and the level of circulation of textbooks increasedgreatly as the number of copies printed rose (Schubring, 1985). Geometrytextbooks, in particular, saw growth in the number of translations of importanttextbooks into several languages. For example, perhaps the most important ofthese being the geometry textbooks by Legendre and Lacroix (Schubring, 1996,367), which were translated from the French to several other languages.

Except in England, the content of geometry textbooks (especially of those for thesenior grades of secondary education) began to deviate more and more fromEuclids Elements, a work that had been the a paradigm of exposition in geometryduring previous centuries. As an example for this development, we can consider

the significant influence Legendres Elements of Geometry (first edition in 1794)(Schubring, 1996, 366) had both on European textbooks and on those of theUnited States during the 19th century. In Legendres book, the study of the circle(3rd Book in Elements) preceded that of parallelograms (2nd Book ofElements).Thus, the adherents of Legendre modified the Euclidean order of presentinggeometry, as they considered the concept of circle simpler and more elementarythan that of the parallelogram (Smith, 1900, 230).

Historical References in Textbooks

Towards the end of 19th century, we see historical referencesbegin to appear moreor less systematically. We use the term historical references to designate theelements of a geometry textbook which refer to the history of geometry, andwhich are written so that omitting them from the textbook would not bedetrimental to its conceptual geometrical content. During the first stage of theirappearance in textbooks, historical references were never an organic part of themain part of text. Instead, they are generally contained either in the booksgeneral introduction, or in the introductions of chapters, or at a chapters close. Ifthey do appear within the text, it is as commentary in brackets and/or small


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The Theorem of Thales 59


point, or even in footnotes (which are usually in smaller point)). As a rule, thesehistorical references refer to the work of mathematicians of antiquity, by namingof theorems the latter are held to have proven, or not to have proven. Sometimes,such references contain a brief summary of the evolution of geometry, or a noterelating to a specific geometrical subject, and to related dates. Such references

were not altogether novel in geometry textbooks, they were present in oldertextbooks as well. However, it was not a recognized technique, but one used byonly a few authors. For example, there are summaries of geometrys evolution intextbooks dating from the 17th century, as in Beaulieu ,1676; Leyboyrn, 1690; LeClerc, 1690 (Kokomoor, 1928, 101). The name Lunes of Hippocrates appears ina textbook of the 17th century2 as well.

We believe that the increased use of historical references in geometry textbooksat the end of 19th century is related to some extent to the substantial advancesmade in the historiography of mathematics after 1870, mainly in England,France, Germany and Italy, in connection with the rise of interest in historical

studies in general (Allman 1877, 160- 161). This led some textbook authors andteachers of mathematics to try to use history in the teaching of mathematics, ina more systematic way than before (Dauben 1999, 110, quoting G. Enestrm).The introduction, in particular, of references to Ancient Greece can be explainedby the rising general interest, in Ancient Greece and Greek mathematical works3

in the course of the 19th century. Some of these references simply consist ofattaching a name to theorems. One of these peculiar historical referencesintroduces the name Theorem of Euclid (only in German textbooks), and another,which we shall study below, introduces the name Theorem of Thales.

Geometrical Achievements Attributed to Thales by Ancient Sources

In Ancient Greek sources, we find: i) five major references to geometricachievements of Thales and ii) some other references concerning his calculationof the height of Egyptian pyramids in Plutarch, Hieronymus the Rhodian, andPliny4. Four of the five major references are found in Proclus. They attribute toThales the following specific theorems: the circle is bisected by its diameter, theangles at the base of an isosceles triangle are equal, the opposite angles are equaland two triangles are equal when they have one side and two adjacent anglesequal (Thomas 2002, 164-167). The other major reference is found in DiogenesLaertiuss biography of Thales, quoting Pamphilas testimony that Thales wasthe first to inscribe in a circle a right-angled triangle (Thomas 2002, 166-169).

Among historians of mathematics, we find diverse views about the correctness ofattributing the above theorems to Thales5 .

In the case of the work (generally ascribed to Thales) which involved measuringthe height of pyramids, historians of the 19th century usually concluded thatThales knew some basic principles of similarity (similar triangles), but theyrarely attributed a specific theorem to him. We know of only one work in thehistory of Mathematics which explicitly mentions two theorems relevant to this


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type of measurement (Ball 1908, 14-16), and these are theorems of EuclidsElements: If a straight line be drawn parallel to one of the sides of a triangle, itwill cut the sides of the triangle proportionally (VI, 2) and: In equiangulartriangles, the sides about the equal angles are proportional (VI, 4) (Heath 1926,194-5, 200-1).

Three historians mention the name Theorem of Thales, but two of these onlymention it to reject it (Loria 1914, 22) (Tannery 1930, 67), and the third, G.Enestrm, voices serious objections to the appropriateness of the name (Enriques1911, 57). The case of D.E. Smith is different. He was active both as a recognizedscholar of the history of mathematics and as a textbook author. It is notable,however, that he never used the name Theorem of Thales in his texts, despite thefact that his textbooks contain historical references to Thales (Wentworth-Smith,c1913, 32).

How the Name Theorem of ThalesEmerged and Became Established

Early attributions of various theorems to Thales

Many years before the name Theorem of Thales emerged, historical referencesappear in geometry textbooks attributing various geometrical achievements toThales. For example, both the Modern Greek translation of Tacquets textbook(Voulgaris, 1805, 25) and its original in Latin (Tacquet, 1722, 20) attribute toThales the calculation of the distance to the inaccessible points by applying thetheorem which states that two triangles are equal when one of their sides andtwo adjacent angles are equal. Also, Benjamin of Lesbos credited Thales with thetheorem about the angle inscribed in a semicircle, as well as the theorem thatopposite angles are equal (Benjamin of Lesbos, 1820, 90, 21). The same author

also mentions that Thales calculated the height of Egyptian pyramids by usingthe proportionality of the sides of similar triangles (Benjamin of Lesbos, 1820, 6).

French textbooks

The name Theorem of Thales first appears in a few French textbooks before the endof 19th century, as early as 1882, (Rouch and Comberousse, 1883, cited in Plane,1995, 79). The name is attributed to the (general) theorem of proportional linesegments or theorem of proportional lines: Des droites parallles dterminent surdes scantes quelconques des segments proportionnels (Parallel lines determineproportional segments on any lines which they cut). However, the very samename is attributed to at least two special cases of the general theorem, as e.g.:

Toute parallle lun des ctes dun triangle partage les deux autres ctes enparties proportionelles (All lines parallel to one of the the sides of a triangle cutthe other two sides proportionally) (Combette, 1882, cited in Plane 1995, 79) andDans le triangle, lgalit des angles entrane la proportionnalit des ctes (twotriangles with equal angles will have proportional sides) (Rouch andComberousse, 1883, cited in Plane 1995, 79). By the 1920s, the name Theorem ofThales was well-established in French geometry textbooks, and mentioned in the


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1925 French curriculum as well (Bkouche, 1995, 9). It also appears in textbooks ofDescriptive Geometry (Cholet- Mineur, 1907- 1908, 315).

Italian textbooks

The theorem of proportional line segments also bears the name ofTheorem of

Thales in Italian geometry textbooks (Faifofer, 1890, 262), at least since 1885(Enrico 1885, 34). It also appears in Italian textbooks on analytic geometry(Enrico, 1885, 34) and analytic-projective geometry (Burali- Forti, 1912, 92).

German textbooks

The name Theorem of Thales is also used in some German textbooks written at theend of 19th century, at least since 1894, but here, it is attributed to a completelydifferent theorem: Der Peripheriewinkel im Halbkreise ist 90 (The angleinscribed in a semicircle is a right angle) (Schwering and Krimphoff, 1894, 53).This name is used for the same theorem (possibly with some variations in thetheorem formulation), in German textbooks during the first decades of the 20th

century. It also shows up in a German encyclopedia of mathematics (Weber-Wellstein- Jacobsthal, 1905, 232).

English and US textbooks

The name Theorem of Thales neither appears in American textbooks, nor in thosepublished in England. In the US, however, we do have references to Thalesconcerning both his geometrical achievements and the measurement methodsattributed to him6. Several of these historical references are due to D.E. Smith(Wentworth/Smith, c1913, 454, 466). In English textbooks, historical referenceswere generally rare during the 19th century.

Other European textbooks

As a consequence of the cultural impact of France and Germany on several otherEuropean countries, the name Theorem of Thales also shows up (with differentmeanings) in those countries textbooks. Thus, the name Theorem of Thalesappears in Spanish (Deruaz-Kogej, 1995, 2390, Belgian (Cambier 1916, 142), andRussian textbooks (Kastanis, 1986, 3) in the same sense as it is used in French andItalian textbooks7. The same name, Theorem of Thales, is employed in Austrian,Hungarian (Howson 1991, 21) and Czech textbooks (Pomylakov 1993, 62)however, but with the meaning attributed to it in German textbooks, rather thanthe French and Italian meaning. Modern Greek textbooks present an exceptionalcase: at first they used the name of the Thales theorem in a sense adhering to thatof German textbooks (Hadjidakis, 1904, 60), and later they switched to the sensegiven to it by French textbooks (Nikolaou, 1927, 128); (Barbastathis 1940, 136).

The Naming of Theorems in the Context of Mathematical Education ofthe 19th Century

The study of how theorems were named in different countries, some of thevariation having been shown above, reflects the different (cultural) conditions


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prevalent in the mathematical education of the countries concerned. Animportant example is provided by France with its time-honored anti-Euclideanoutlook (Schubring, 1996, 377; Cajori, 1910, 182). This viewpoint resulted in atradition which caused the French order of geometry subject-matter to deviatefrom that of Euclid. For example, in Euclid, the theorem of the square of the

hypotenuse (I, 47) precedes the theory of proportions (5th

book ofElements) just aswell as the theorem corresponding to that of proportional lines (VI, 2). In Frenchtextbooks, this order had been reversed since P. Ramuss era. This reversal wasnot followed by German textbooks until the beginning of 20th century8. Italiantextbooks (after 1866) retained the Euclidean order presenting geometry subjectsbecause, at that time, Euclids Elements hadbeen adopted as the official textbookin Italian schools (Schubring, 1996, 377-378; Cajori, 1910, 191).

In the meantime, the theorem of proportional lines had risen to a prominentposition in French textbooks, because of the new developments in (academic)mathematical research in geometry. One of these developments which following

the the works of G. Desargues, B. Pascal, La Hire (1685), Carnot (Coolidge, 1934,219-220), was due to the work of J.V. Poncelet (of 1813, published in 1822)marked the beginning of Projective Geometry and Affine Geometry. A notionmaintaining a key position in projective geometry is harmonic separation(Coolidge, 1934, 220), which refers to an invariant ratio of particular linesegments and thus relates to the theorem of proportional lines. This theorem alsorelates to the new subject of affine geometry because the theorem of proportionallines implies preservation of ratios between collinear line segments which is a keynotion in affine geometry. These new structural mathematical developmentshad in some sense been institutionalized in the teaching of geometry by the end of19th century9. Simultaneously, historical interest in Thales as a mathematician

grew in the 19th century, this interest possibly being one of the reasons why thename Theorem of Thalesbecame the generally accepted designation for the abovetheorem10.

The same situation is all the more true for Italian textbooks. Although Italiantextbooks maintained the Euclidean order of presenting theorems, Italianauthors, particularly L. Cremona, adopted certain elements from projectivegeometry (Cajori, 1910, 191). Some Italian authors, like R. de Paolis, 1884(Candido, 1899, 204) also tried to blend elements from plane geometry and thegeometry of solids, again by drawing on ideas taken from projective geometry.This helped create a preference in Italian authors for the theorem concerning

proportional line segments, which they attributed to Thales.The case of German textbooks is different, since the Germans did not change theEuclidean presentation. Note for example that they showed strong preference forthe theorem concerning the square of the hypotenuse, the so-called PythagoreanTheorem11. This preference exemplifies a school geometry which was morealigned to Euclid, and thus lacked any conceptual association with projective oraffine geometry. (However, this is not to suggest that academic research in


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Germany did not contribute to the development of projective geometry after1830).

It would appear that names are assigned to theorems considered essential andsignificant for school geometry12. Thus, French textbooks chose to ascribe thename Theorem of Thales to a theorem which was essential for the modern viewof geometry (projective, affine), while German textbooks chose to ascribe thename to a theorem important for classic Euclidean geometry, but containing alarge number of modern elements (algebraic calculations). Modern Europeantextbooks in other countries were far removed from Euclidean geometry, andsaw the necessity of stressing the importance of theorems essential to newdevelopments in geometry.

Towards a Didactical Explanation of the Name

Our study pertains to those contents of school textbooks which do not referdirectly to mathematical concepts, but to the historical origin of those concepts.

Textbook authors were always looking for ways to emphasize the importance ofcertain theorems in the curricula. Naming a theorem is a symbolic act which goesfar beyond presenting a simple historical landmark. After having introduced thehistory of a concept or theorem which the textbook authors had emphasized intheir text, they tried to profit from history. Naming the theorem after anundisputed, time-honored authority to which it could be historically tracedfurther authenticates it and establishes its importance.

In the first phase of this process, we find some simple historical references toThales, without namingof theorems after him. A French textbook of 1866(Rouch and Comberousse, 1866, v) for example, attributes a theory of similar

triangles to Thales, while a German textbook of 1875 (Kruse, 1875, 18, 64)attributes two theorems (opposite angles are equal; in a semicircle, the angle is aright angle) to the same Thales. In a later stage, the historical references to Thalesevolve into naming of the corresponding theorems after him, and into using of hisname not in footnotes or asides, but within the main body of text. Establishingthe name Theorem of Thales in this later stage served a didactical need by usingthe history of geometry to underpin the particular didactical-ideologicalinterpretation the textbook authors favored. We are thus led to speak, in analogyto didactical transposition (Chevallard and Joshua, 1982) about a didacticalreconstruction of History of geometry, i.e. a reconstruction of history of geometryfor didactical requirements.

The authors of school textbooks are as a rule not historians of mathematics;.Rather, they are teachers who evidently desire to profit didactically from thehistory of geometry. They seem to know about Thales from a historical source,and try to establish a direct link between Thales and a theorem already present inschool geometry. There is no mention of sources from antiquity at all uponintroducing attributions of theorems in textbooks of school geometry,particularly not in connection with the name Theorem of Thales. What we find are


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only different choices from historical sources among these theorems ascribed toThales, with no discussion or historical argumentation at all. Historians ofmathematics proceed in a totally different way. For example, they argue at lengthwhether the ancient sources about Thales are valid, or not, trying to see if Thalesindeed gave proofs, and if he employed some general method.

We consider that the concept of didactical reconstruction offers a key tounderstanding how textbook authors used the history of mathematics. Theyfocused on its didactical advantages, treating history as one educational toolamong others. Here, we have shown that to emphasize a theorems value, theyattributed it to a famous mathematician of antiquity (Thales). In other cases, theyselected topics or subjects from the history of mathematics according to theirown national and cultural tradition. We believe that there is more to investigateand learn in mathematics education by examining historical references in schooltextbooks. The name Theorem of Thales is only a particular, though telling,example in this direction.

Acknowledgements

This is a revised and in-depth version of a presentation that we made at theinternational conferenceHistory and Pedagogy of Mathmeatics, Uppsala 12-17 July2004, Sweden. We thank the participants of the conference, as well as thereviewers of this text, for their valuable comments and contributions.

Notes

1. M. Gebhardts work, Die Geschichte der Mathematik im mathematischen Unterrichte, 1912,remains an exception, for a description of the book see (Furinghetti, 2001, 1).

2. The name Lunes de Hippocrate de Scio appears in the textbook Elemens de Geometrie of theJesuit Pardies (1676), as well as in Tacquets textbook (1745) (Lietzmann, 1912, 35). Thename appears also in the Modern Greek translation of the Tacquet textbook as conjugatelunes of Hippocrates of Chios (Boulgaris, 1805, 296, 302), This translation draws on the 1710edition of Tacquets textbook (Karas, 1993, 70).

3. There were many editions of ancient Greek mathematical works at the end of 19thcentury, for examples see (Allman, 1877, 161).

4. There is an exhaustive presentation of the literature about Thales and geometry in (Tezas,1990, 61-103).

5. We refer to three of the most important of these historians, (Cantor 1907, 134-147; Heath1921, 128-137; Tannery 1887, 81-94). At this point, we should like to make a distinctionbetween the research works in the history of mathematics, and the works addressed to awider audience (popular works). In any case, we have not found a book on history ofmathematics written only for the wider audience in the second half of the 19th century.

6. As examples, we may mention the calculationthe height of Pyramids, and measuringinaccessible points (Betz and Webb, c1912, 281, 68). In another textbook, among other


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achievements, the theorem that every diameter bisects a circle is attributed to Thales(Fletcher, 1911, 496).

7. In Russian textbooks, we read the following theorem as Theorem of Thales: if twotransversals are given and three or more parallel lines intercept congruent segments onone of them, then they also intercept congruent segments on the other one. In addition,

the theorem of French and Italian textbooks was sometimes named Generalized ThalesTheorem (information by professor G.Schubring). In later Greek education, the theoremabout transversals had the name Lemma of Thales (information by professor T. Patronis).

8. This change of order had an impact on the proof of the corresponding theorems (I, 47 andVI, 2). We observe that the complete proof of the theorem VI, 2 and the analogoustheorems of section 5.2 requires the theory of proportion (book V), or, in general, sometheory of irrational numbers (limits or Dedekind cuts). The historical stages in thedevelopment of the proof of the theorem of proportional lines from Euclid to Dedekindmay be found in Bkouche, 1995, but textbooks contain a great variety of incompleteproofs, which have their own interest and their own history. We note that textbook

authors do not relate the name of a theorem to its proof, otherwise they would notattribute to Thales (a beginner of geometry) a theorem which requires so difficult a proof.

9. According to Patronis (2002, 68), institutionalization is, first and foremost, a symbolic actof showing what is important and respectable within human society, or within a contextof social activity. In this sense, institutionalization in the context of (mathematical)education uses (names of) historical figures as Thales, Pythagoras, and sometimes alsoEuclid to assign a status to the subject taught.

10. More generally, there was a growing historical interest in early Greek mathematicsfostered by C. A. Bretschneiders Die Geometrie und die Geometer vor Euklides, 1870, see(Allman, 1877, 161).

11. There are at least four books in German which study this theorem: J. Hoffmann (1819), J.Wipper (1880), H.A. Naber (1908), of course, Lietzmann, 1912. Lietzmann refers toHoffmann, Wipper and Naber at the page 70).

12. This is also the case for the name Pythagorean Theorem, as well as for the name ofTheoremof Hippocrates of Chios in Modern Greek textbooks of Geometry (Patsopoulos, 2003, 577).

References

Allman, G.J. (1877). Greek Geometry from Thales to Euclid. Hermathena, vol. III(5). Dublin: Printed at the University Press by Ponsonby and Murphy.

Ball, W. (1908). A short account of the History of Mathematics. (1st edition 1888).Reprinted in New York: Dover Publications, Inc.

Barbastathis, X.A. (1940). Theoretical Geometry, For 4th, 5th, 5th classes of new typeGymnasium. Athens: Organization of Edition of Textbooks. [in Greek]

Benjamin of Lesbos (1820). Elements Of Geometry of Euclid. Venice: IoannisSneirer.[in Greek]


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Betz, W., Webb, H.E. (circa 1912). Geometry, with the editorial cooperation ofP.F.Smith. Boston- New York- Chicago-London- Dallas- Columbus- SanFrancisco: Ginn Company.

Bkouche, R. (1995). Variations sur les liens entre le gomtrique et le numrique:Autour du thorme de Thals. InAutour de Thals (pp.7-67). Comission Inter-IREM Premier Cycle.Burali- Forti, C. (1912). Corso di Geometria Analitico-Proiettiva per gli allievi dellaAccademia Militare. Torino: Casa Editrice G.B. Petrini di Giovanni Gallizio.

Cajori, F. (1910). Attempts made during the Eighteenth and Nineteenth centuriesto reform the teaching of Geometry. American Mathematical Monthly, 17(10),181-201.

Cambier A. (1916). lments de Gomtrie. (dition revue et complete par O.Lambot). Bruxelles : Maison d dition A. de Boeck.

Candido, G. (1899). Sur la fusion de la Planimtrie et de la Stromtrie dans lenseignement de la Gomtrie lmentaire en Italie. L EnseignementMathmatique, 1re Anne, 204-215.

Cantor, M. (1907). Vorlesungen ber Geschichte der Mathematik. (Erster band,Nachdruck der dritten Auflage von 1907). Stuttgart: B.G. Teubner. Reprinted in1965, New York: Johnson Reprint Corporation.

Chevallard, Y., Joshua M.-A. (1982). Un exemple d analyse de la TranspositionDidactique, La notion de distance. Recherches en Didactique de Mathmatiques, 3 (2),159-239.

Cholet, T., Mineur P. (1907-1908). Trait de Gomtrie descriptive. Paris : Vuibert et

Nony.Coolidge, J.L. (1934). The rise and fall of Projective Geometry. AmericanMathematical Monthly, 41 (4), 217-228.

Dauben, J.W. (1999). Historia Mathematica: 25 Years. Context and Content.Historia Mathematica, 26, 1-28.

Deruaz H., Kogej N. (1995). Le thorme de Thals : comment est-il enseign enEurope?. InAutour de Thals (pp.233-243). Comission Inter-IREM Premier Cycle.

Enrico, O. d. (1885). Teoria analitica delle forme geometriche fondamentali, Lezionidate nella Regia Universit di Torino. Torino: E. Loescher.

Enriques F. (1911). Principes de Geometrie. In Molk J. (ed.), Encyclopdie desScience Mathmatiques Pures et Appliques (dition Franaise, Tome III, 2e volume).Paris: Gauthier- Villars. Reprinted in 1991, Paris: ditions Jacques Gabay, 1-147.

Faifofer, A. (1890). Elementi di Geometria ad uso degl instituti tecnici (1o biennio) e deilicei. (7th edition). Venezia: Tipografia Emiliana.

Fletcher, D. (1911). Plane and Solid Geometry. New York: Charles E. Merill.


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Furinghetti, F. (2001). A century ago. History and pedagogy of MathematicsNewsletter,48, 1.

Hadjidakis, I.N. (1904). Elements of Geometry,For the pupils of Gymnasium. (8th

edition). Athens: Editions of P.D. Sakellariou. [in Greek]

Heath, T.L. (1921).A History of Greek Mathematics.Oxford: Clarendon Press.Heath, T.L. (1926). The thirteen books of Euclids Elements, translated from the text ofHeiberg. (2nd vol., 2nd edition, 1st edition 1908). Reprinted in New York: DoverPublications Inc.

Howson, G. (1991). National Curricula in Mathematics. Avon: The MathematicalAssociation, Bath Press.

Karas, G. (1993). Science under Othoman Empire, Manuscripts and books, 1st vol.:Mathematics. Athens: Bookshop of Hestia. [in Greek]

Kastanis, N. (1986). A case of historical confusion in Geometry textbooks.

Newsletter of the Group for the History of Mathematics, 2, 1986, 3-4 [in Greek].Kokomoor, F.W. (1928). The teaching of Elementary Geometry in the Seventeenthcentury. Isis, 11 (1), 85-110.

Kruse, F. (1875). Elemente der Geometrie. Berlin: Weidmann.

Lietzmann W. (1912). Der Pythagoreische Lehrasatz. Leipzig-Berlin: Druck undVerlag von B.G. Teubner.

Loria, G. (1914). Le Scienze esatte nell antica Grecia. (2nd ed. Totalmenteriveduta). Milano: U.Hoepli.

Nikolaou, N.D. (1927), Elementary Geometry. For use of the pupils ofGymnasium and Lyceum. Athens: Publishing House of D.N. Tzaka- S.Delagrammatica & Co.[in Greek]

Patronis, A. (2002). Ideology and institutionalization of knowledge: Questionsthat History of Mathematics proposes in Mathematical Education and vice versa.In Proceedings of 19th Panellenic Conference of Mathematical Education (pp. 63-72),Komotini: Greek Mathematical Society. [in Greek]

Patsopoulos, D. (2003). Theorem of Hippocrates of Chios: Constructing (thename of) a theorem for modern Greek textbooks of Geometry. In Gagatis A.,Papastavridis S. (eds), Proceedings of 3rd Mediterranean Conference on Mathematicaleducation (pp. 573-579) Athens: Greek Mathematical Society, CyprusMathematical Society.

Plane, H. (1995). Une invention franais du XXe sicle : le thorme de Thals. InAutour de Thals (pp.68-85. Comission Inter-IREM Premier Cycle.

Pomylakov, E. (1993).Matematika, Planimetrie, Pro Gymnzia. Prometheus.

Rouch E., Comberousse C. de (1866). Trait de Gomtrie lmentaire. Paris:Gauthier-Villars.


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Schubring, G. (1985). Essais sur lHistoire de l Enseignement des Mathematiquesparticulirement en France et en Prusse. Recherches en Didactiques desMathematiques,5(3), 343-385.

Schubring, G. (1996). Changing cultural and epistemological views onmathematics and different institutional contextes in nineteenth-century Europe.In Goldsterin C., Gray J., Ritter J., (eds), L Europe Mathematique- Mythes, histoires,identits.Mathematical Europe- Myth, History, Identity. Paris: ditions de la Maissondes Sciences de lHomme, 363-388.

Schwering K., Krimphoff W. (1894). Anfangsgrde der ebenen Geometrie. Freiburgim Breigsgau: Herdersche Verlagshandlung.

Smith, D.E. (1900). The teaching of elementary Mathematics. London- New York:Macmillan & Co, Ltd.

Tacquet, A. (1722). Elementa Euclidea Geometriae planae et solidae et selecta exArchim ede theoremata, addidit G.Whinston. Cantabrigiae, Impensis

Corn.Crownfield.Tannery, P. (1887). La Gomtrie Grecque. Paris : Gauthier-Villars. Reprinted in1988, Paris : ditions Jacques Gabay.

Tannery, P. (1930). Pour lHistoire de la Science Hellne, De Thals a Empdocle. (2e

dition). Paris: Gauthier- Villars et Cie. Reprinted in 1990, Paris :ditions JacquesGabay.

Tezas, C.A. (1990). Thales the Milesian and the beginning of the Science: The road toPhilosophy, Ioannina: University of Ioannina.

Thomas, I. (2002). Selections illustrating the History of Greek mathematics. 1st vol.

(1st edition 1939). Loeb Classsical Library. Cambridge, Massachussets London,England:Harvard University Press.

Voulgaris, E. (1805). Elements of Geometry of A. Tacquet, with notes by W. Whinston.Venice: Georgios Vendotis. [in Greek]

Weber, H., Wellstein, J., Jacobsthal, W. (1905). Encyclopdie der Elementar-Mathematik, zweiter band: Encyclopdie der Elementaren Geometrie. Leipzig: Druckund Verlag von B.G. Teubner.

Wentworth, G., Smith, D.E. (circa 1913). Plane and Solid Geometry, Boston- NewYork- Chicago-London- Dallas- Columbus- San Francisco: Ginn and Company.

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